Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphs

IntroductionMatching

Our Contribution

Space Efficient Approximation Scheme forMaximum Matching in Sparse Graphs

Samir Datta Raghav Kulkarni Anish Mukherjee

Chennai Mathematical Institute

NMI Workshop on Complexity Theory, IIT Gandhinagar

November 04, 2016

Samir Datta Raghav Kulkarni Anish Mukherjee Space Efficient Approximation Scheme for Maximum Matching in Sparse Graphs


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Overview

1 Introduction

2 Matching

3 Our Contribution



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Baker’s Algorithm

Theorem (Baker ’83, Informal)

A class of problems (many of which are NP-Hard in general) canbe approximated arbitrarily close to the optimal value in linear timefor planar graphs.

Example

Includes problems like

maximum independent set

partition into triangles

minimum vertex-cover

minimum dominating set

... and any MSO definable properties



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Baker’s Algorithm



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Baker’s Algorithm



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Baker’s Algorithm

Basic Idea

Partition the vertices into breadth-first search levels

Decompose the graph into successive width-k slices bydeleting levels congruent to i mod k



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Baker’s Algorithm

Basic Idea





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Baker’s Algorithm

Basic Idea





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Baker’s Algorithm

Basic Idea





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Baker’s Algorithm

Basic Idea



Resulting components have treewidth 3k − 1 [Boadlander]



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Baker’s Algorithm

Basic Idea






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Baker’s Algorithm

Basic Idea






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Baker’s Algorithm

Basic Idea




Solve the problem optimally in each partition in linear time

Union of solutions in all components is within (1− 1/k) OPT



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Baker’s Algorithm

Basic Idea




Solve the problem optimally in each partition in linear time

Union of solutions in all components is within (1− 1/k) OPT



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Baker’s Algorithm II

But here we are interested in space efficient algorithms,namely algorithms running in Logspace

EJT gives an algorithm for Courcelle’s theorem in Logspace

But for the first part we need to compute distance

Distance is NL-Complete in general undirected graphs and inUL ∩ co-UL for planar graphs.

And these classes are not believed to be in Logspace.

Question

Can we get away without using distance ? Not yet



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Question




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Question

Can we get away without using distance ?

Not yet



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Question




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Overview

1 Introduction

2 Matching

3 Our Contribution



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Matching

Matching

A matching M ⊆ E is a set of independent edges

A matching M is called perfect if M covers all vertices of G

M of maximum size is called maximum matching

Augmenting Paths

In an alternating path the edges alternate between M andE \MAn alternating path P is augmenting if P begins and ends atunmatched vertices, that is, M ⊕P = (M \P )∪ (P \M) is amatching with cardinality |M |+ 1.



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Matching

Matching




Augmenting Paths




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Matching

Matching




Augmenting Paths

In an alternating path the edges alternate between M andE \M

An alternating path P is augmenting if P begins and ends atunmatched vertices, that is, M ⊕P = (M \P )∪ (P \M) is amatching with cardinality |M |+ 1.



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Matching

Matching




Augmenting Paths




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Matching

Edmond’s blossom algorithm for maximum matching was oneof the first examples of a non-trivial polynomial time algorithm

Valiant’s #P-hardness for counting perfect matchings gavesurprising insights into the counting complexity classes

The RNC bound for maximum matching has yielded powerfultools, such as the isolating lemma [MVV87]

Bipartite Perfect Matching is in quasi-NC [FGT16]

The best hardness known is NL-hardness [CSV84]



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Matching in Planar Graphs

Counting perfect matchings in planar graphs is in NC [Vaz88]

Only the bipartite planar case is known to be in NC for findinga perfect matching [MN95]

Open Problem

Is the construction version in general planar graphs in NC ?

Computing a maximum matching for bipartite planar graphs isshown to be in NC [Hoang]

Only L-hardness is known for planar graphs [DKLM10].



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Open Problem






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Time-Space Tradeoff

Removing non-determinism even for planar reachability leadsto either a quasi-polynomial time blow-up or need large space(O(√n)) [INPVW13, AKNW14]

For general graphs it is even worse, with O(n/2√logn) space

and polynomial time [BBRS]

Previous Results

Approximating maximum matching has been considered bothin time and parallel complexity model

Linear-time [DP14] and NC [HV06] approximation scheme arethe best known complexity bounds here

But work on space efficient approximation seems limited.



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Time-Space Tradeoff




Previous Results






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Time-Space Tradeoff




Previous Results






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Time-Space Tradeoff




Previous Results






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Overview

1 Introduction

2 Matching

3 Our Contribution



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Results

Theorem

Given a planar graph and any fixed ε > 0, we can find a (1− ε)factor approximation to the maximum matching in Logspace.

This result extends to many other sparse graph classes

Some of our ideas are similar to the classical algorithm ofHopcroft-Karp for maximum matching in bipartite graphs

But we consider graphs which are not necessarily bipartite

Our algorithm trades off Logspace and non-bipartiteness forapproximation and sparsity

Solve by reducing it to bounded degree graphs suitably.



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Results

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Results

Theorem

Let G be a graph with degrees bounded by a constant d then forany fixed ε > 0, we can find a (1− ε) factor approximation to themaximum matching in Logspace.

The main fact we use here is that any bounded degree graphsalways contains a linear size matching

Many planar graph classes, such as 3-connected planargraphs, are known to be containing a large matching

In fact our algorithm works for any recursively sparse graphcontaining a large matching.



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Results

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A Brief Idea

1 Consider short augmenting paths. In a bounded degree graph,there exist linearly many short augmenting paths

2 Pick a large subset of non-intersecting augmenting paths i.efind a large independent set of in Logspace

3 To convert a planar graph to a bounded degree graph wedelete high degree vertices

4 The number of such vertices is small though possibly stilllinear in the graph size

5 Remove small number of vertices and edges to transform thegraph down to one containing a linear sized matching.



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A Brief Idea








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Bounded degree graphs I

We deal with augmenting paths of length at most 2k + 1

Such paths can be found in Logspace by say exhaustivelylisting all (2k + 1)-tuples of vertices using L-transducers

If |M | differs significantly from |Mopt| then we show thatthere are many short augmenting paths

Lemma

If |M | < (1− 3k )|Mopt| for some k then there are at least

3|Mopt|/2k augmenting paths consisting of at most 2k + 1 edges.

Form an intersection graph of these short augmenting pathsby making two paths adjacent if they have a vertex in common



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Maximum matching in bounded degree graphs II

Lemma

A β-factor approximation to the maximum independent set can becomputed in Logspace

Colour the paths and the largest colour class works

As the degree is bounded by some D, find at most D disjointforests that partition the edge set

Can be done using Reingold’s algorithm for connectivity

Colour each forest with 2 colours and it gives D bit colours toevery node

This yields a 2D i.e. constant colouring of the graph.



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Theorem

In a bounded degree graph for any fixed ε > 0, we can find a(1− ε) factor approximation to the maximum matching in L.

Previous lemma yields large fraction of short paths,augmentable in parallel

A L-transducer can do the augmentation and we chain(1− 3/k)2k/β such transducers

At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.



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At each step we increase the matching size by an additiveterm of |Mopt|/(2k/β)

After k rounds the ratio would be at least (1− 3/k) ≥ 1− ε.



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Algorithm 1

1 Fix integer k =⌈3ε

⌉.

2 Construct the intersection graph of augmenting paths oflength at most 2k + 1 in G.

3 Let the graph be H with maximum degree≤ D = (2k + 1)2d2k+1

4 Find at most D disjoint forests that partition the edge set.

5 Colour each forest with 2 colours, giving D bit colours toevery node

6 Augment the vertex disjoint augmenting paths in parallel

7 Add the new matching to M

8 Return M



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Planar maximum matching

Definition

A graph is tame if all pairs of vertices (a, b) which are endpoints ofa even length isolated path, support at most two of them.

This can be ensured by deleting a set of edges E′ from G

Lemma

The size of the maximum matching in G \ E′ is the same as in G.

Main Lemma

A tame planar graph has a linear sized maximum matching.



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Definition



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Main Lemma




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Planar maximum matching: tame graphs

One of the following is true :

Total length of long isolated paths in G′ is large enough

We can transform the graph by case analysis to a minimumdegree 3 planar graph

Lemma

A graph in which the total length of isolated paths is N has amatching of size at least N/4.

Lemma

A min degree 3 planar graph has a matching of size at least n/140.



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Planar maximum matching III

Theorem

There is a LSAS for maximum matching in planar graphs.

proof

Tame the graph G to G′ preserving the maximum matchingsize. Suppose there are least αn matching edges in G′

Delete vertices of degree more than d from G′ which removesat most 6n/d many matching edges

So we have a (α− 6/d)n sized matching remaining

Taking d = 122α−ε reduces the problem to find a (1− ε/2)

factor approximation algorithm for bounded degree graphs.



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Conclusion

We showed that maximum matching can be approximated toany arbitrary constant factor for bounded degree graphs

For planar graphs we require only the following properties:

Sparsity: The average degree is bounded by 6.Bipartite sparsity: Even lower, i.e 4.Min-degree: The minimum degree is at least 3

So can be extended many other classes of sparse graphs

bounded genus graphs,k-page graphs,1-planar graphs, k-Apex graphs etcrecursively sparse graph containing a linear size matching.



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Conclusion








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Open Problems

Baker’s Theorem in Logspace ?

Devise an LSAS for maximum matching in general graphs

or at least in arbitrary sparse graphs

Lower bounds in the context of approximation ?

Currently we do not know of any non-trivial, evenTC0-hardness results for approximation to any factor.



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Open Problems








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Open Problems








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Thank You



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Packing complex patterns ?

H-Matching

Pack disjoint copies of a fixed graph H

Maximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.

Approximation and hardness is known for some restricted cases

We give LSAS for graphs with a small balanced separator,

for packing any fixed graph H when degrees are bounded

Otherwise, Packing some special class of patterns

As before, the idea is to delete high degree vertices

and tame the graph by removing some forbidden patterns



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H-Matching

Pack disjoint copies of a fixed graph HMaximum planar H-matching is NP-Complete for any Hcontaining at least three nodes.









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Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphs